Optimal. Leaf size=98 \[ \frac {a^{4 k x}}{4 k \log (a)}+\frac {a^{4 l x}}{4 l \log (a)}+\frac {3 a^{2 (k+l) x}}{(k+l) \log (a)}-\frac {4 a^{(3 k+l) x}}{(3 k+l) \log (a)}-\frac {4 a^{(k+3 l) x}}{(k+3 l) \log (a)} \]
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Rubi [A]
time = 0.07, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {6874, 2225}
\begin {gather*} \frac {3 a^{2 x (k+l)}}{\log (a) (k+l)}-\frac {4 a^{x (3 k+l)}}{\log (a) (3 k+l)}-\frac {4 a^{x (k+3 l)}}{\log (a) (k+3 l)}+\frac {a^{4 k x}}{4 k \log (a)}+\frac {a^{4 l x}}{4 l \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2225
Rule 6874
Rubi steps
\begin {align*} \int \left (a^{k x}-a^{l x}\right )^4 \, dx &=\frac {\text {Subst}\left (\int \left (e^{k x}-e^{l x}\right )^4 \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {Subst}\left (\int \left (e^{4 k x}+e^{4 l x}+6 e^{2 (k+l) x}-4 e^{(3 k+l) x}-4 e^{(k+3 l) x}\right ) \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {\text {Subst}\left (\int e^{4 k x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {\text {Subst}\left (\int e^{4 l x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac {4 \text {Subst}\left (\int e^{(3 k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}-\frac {4 \text {Subst}\left (\int e^{(k+3 l) x} \, dx,x,x \log (a)\right )}{\log (a)}+\frac {6 \text {Subst}\left (\int e^{2 (k+l) x} \, dx,x,x \log (a)\right )}{\log (a)}\\ &=\frac {a^{4 k x}}{4 k \log (a)}+\frac {a^{4 l x}}{4 l \log (a)}+\frac {3 a^{2 (k+l) x}}{(k+l) \log (a)}-\frac {4 a^{(3 k+l) x}}{(3 k+l) \log (a)}-\frac {4 a^{(k+3 l) x}}{(k+3 l) \log (a)}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 80, normalized size = 0.82 \begin {gather*} \frac {\frac {a^{4 k x}}{k}+\frac {a^{4 l x}}{l}+\frac {12 a^{2 (k+l) x}}{k+l}-\frac {16 a^{(3 k+l) x}}{3 k+l}-\frac {16 a^{(k+3 l) x}}{k+3 l}}{4 \log (a)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 109, normalized size = 1.11
method | result | size |
risch | \(\frac {a^{4 k x}}{4 k \ln \left (a \right )}-\frac {4 a^{3 k x} a^{l x}}{\ln \left (a \right ) \left (3 k +l \right )}+\frac {3 a^{2 k x} a^{2 l x}}{\ln \left (a \right ) \left (k +l \right )}-\frac {4 a^{k x} a^{3 l x}}{\ln \left (a \right ) \left (k +3 l \right )}+\frac {a^{4 l x}}{4 l \ln \left (a \right )}\) | \(109\) |
meijerg | \(-\frac {1-{\mathrm e}^{4 k x \ln \left (a \right )}}{4 k \ln \left (a \right )}+\frac {4-4 \,{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2 k}{l \left (1+\frac {k}{l}\right )}\right )}}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2 k}{l \left (1+\frac {k}{l}\right )}\right )}-\frac {3 \left (1-{\mathrm e}^{2 x l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}\right )}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right )}+\frac {4-4 \,{\mathrm e}^{x l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2}{1+\frac {k}{l}}\right )}}{l \ln \left (a \right ) \left (1+\frac {k}{l}\right ) \left (1+\frac {2}{1+\frac {k}{l}}\right )}-\frac {1-{\mathrm e}^{4 l x \ln \left (a \right )}}{4 l \ln \left (a \right )}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.76, size = 99, normalized size = 1.01 \begin {gather*} -\frac {4 \, a^{3 \, k x + l x}}{{\left (3 \, k + l\right )} \log \left (a\right )} - \frac {4 \, a^{k x + 3 \, l x}}{{\left (k + 3 \, l\right )} \log \left (a\right )} + \frac {3 \, a^{2 \, k x + 2 \, l x}}{{\left (k + l\right )} \log \left (a\right )} + \frac {a^{4 \, k x}}{4 \, k \log \left (a\right )} + \frac {a^{4 \, l x}}{4 \, l \log \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 207 vs.
\(2 (94) = 188\).
time = 0.51, size = 207, normalized size = 2.11 \begin {gather*} -\frac {16 \, {\left (3 \, k^{3} l + 4 \, k^{2} l^{2} + k l^{3}\right )} a^{k x} a^{3 \, l x} - 12 \, {\left (3 \, k^{3} l + 10 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{2 \, k x} a^{2 \, l x} + 16 \, {\left (k^{3} l + 4 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{3 \, k x} a^{l x} - {\left (3 \, k^{3} l + 13 \, k^{2} l^{2} + 13 \, k l^{3} + 3 \, l^{4}\right )} a^{4 \, k x} - {\left (3 \, k^{4} + 13 \, k^{3} l + 13 \, k^{2} l^{2} + 3 \, k l^{3}\right )} a^{4 \, l x}}{4 \, {\left (3 \, k^{4} l + 13 \, k^{3} l^{2} + 13 \, k^{2} l^{3} + 3 \, k l^{4}\right )} \log \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1348 vs.
\(2 (82) = 164\).
time = 20.60, size = 1348, normalized size = 13.76 \begin {gather*} \begin {cases} 0 & \text {for}\: a = 1 \wedge \left (a = 1 \vee k = 0\right ) \wedge \left (a = 1 \vee l = 0\right ) \\\frac {a^{4 l x}}{4 l \log {\left (a \right )}} - \frac {4 a^{3 l x}}{3 l \log {\left (a \right )}} + \frac {3 a^{2 l x}}{l \log {\left (a \right )}} - \frac {4 a^{l x}}{l \log {\left (a \right )}} + x & \text {for}\: k = 0 \\\frac {a^{4 l x}}{4 l \log {\left (a \right )}} - 4 x - \frac {3 a^{- 4 l x}}{2 l \log {\left (a \right )}} + \frac {a^{- 8 l x}}{2 l \log {\left (a \right )}} - \frac {a^{- 12 l x}}{12 l \log {\left (a \right )}} & \text {for}\: k = - 3 l \\\frac {a^{4 l x}}{4 l \log {\left (a \right )}} - \frac {2 a^{2 l x}}{l \log {\left (a \right )}} + 6 x + \frac {2 a^{- 2 l x}}{l \log {\left (a \right )}} - \frac {a^{- 4 l x}}{4 l \log {\left (a \right )}} & \text {for}\: k = - l \\- \frac {3 a^{\frac {8 l x}{3}}}{2 l \log {\left (a \right )}} + \frac {9 a^{\frac {4 l x}{3}}}{2 l \log {\left (a \right )}} + \frac {a^{4 l x}}{4 l \log {\left (a \right )}} - 4 x - \frac {3 a^{- \frac {4 l x}{3}}}{4 l \log {\left (a \right )}} & \text {for}\: k = - \frac {l}{3} \\\frac {a^{4 k x}}{4 k \log {\left (a \right )}} - \frac {4 a^{3 k x}}{3 k \log {\left (a \right )}} + \frac {3 a^{2 k x}}{k \log {\left (a \right )}} - \frac {4 a^{k x}}{k \log {\left (a \right )}} + x & \text {for}\: l = 0 \\\frac {3 a^{4 k x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 k x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 k x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {3 a^{4 k x} l^{4}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} - \frac {16 a^{3 k x} a^{l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} - \frac {64 a^{3 k x} a^{l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} - \frac {48 a^{3 k x} a^{l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {36 a^{2 k x} a^{2 l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {120 a^{2 k x} a^{2 l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {36 a^{2 k x} a^{2 l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} - \frac {48 a^{k x} a^{3 l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} - \frac {64 a^{k x} a^{3 l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} - \frac {16 a^{k x} a^{3 l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {3 a^{4 l x} k^{4}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 l x} k^{3} l}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {13 a^{4 l x} k^{2} l^{2}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} + \frac {3 a^{4 l x} k l^{3}}{12 k^{4} l \log {\left (a \right )} + 52 k^{3} l^{2} \log {\left (a \right )} + 52 k^{2} l^{3} \log {\left (a \right )} + 12 k l^{4} \log {\left (a \right )}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 0.86, size = 1359, normalized size = 13.87 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.35, size = 106, normalized size = 1.08 \begin {gather*} \frac {3\,a^{2\,k\,x}\,a^{2\,l\,x}}{k\,\ln \left (a\right )+l\,\ln \left (a\right )}-\frac {4\,a^{k\,x}\,a^{3\,l\,x}}{k\,\ln \left (a\right )+3\,l\,\ln \left (a\right )}-\frac {4\,a^{3\,k\,x}\,a^{l\,x}}{3\,k\,\ln \left (a\right )+l\,\ln \left (a\right )}+\frac {a^{4\,k\,x}}{4\,k\,\ln \left (a\right )}+\frac {a^{4\,l\,x}}{4\,l\,\ln \left (a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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